The Power of Data Reduction for Matching

TL;DR

This paper explores how linear-time data reduction techniques can significantly speed up the process of finding maximum matchings in graphs, applicable across various solution strategies.

Contribution

It introduces systematic, linear-time kernelization methods for graph matching problems, applicable to both general and bipartite graphs, enhancing preprocessing efficiency.

Findings

Data reduction algorithms are compatible with all solution strategies.

Kernelization significantly reduces problem size in linear or near-linear time.

Applicable to both exact and approximate matching algorithms.

Abstract

Finding maximum-cardinality matchings in undirected graphs is arguably one of the most central graph primitives. For $m$-edge and $n$-vertex graphs, it is well-known to be solvable in $O(m\sqrt{n})$ time; however, for several applications this running time is still too slow. We investigate how linear-time (and almost linear-time) data reduction (used as preprocessing) can alleviate the situation. More specifically, we focus on (almost) linear-time kernelization. We start a deeper and systematic study both for general graphs and for bipartite graphs. Our data reduction algorithms easily comply (in form of preprocessing) with every solution strategy (exact, approximate, heuristic), thus making them attractive in various settings.